Let's denote the width of the rectangle as $w$. According to the problem, the length $l$ is 4 times the width, so:

$l = 4w$

The area $A$ of the rectangle is given as 44 cm², and the area of a rectangle is calculated as:

$A = l \times w$

Substituting the values:

$44 = 4w \times w$

$44 = 4w^2$

To find $w$, divide both sides by 4:

$w^2 = \frac{44}{4} = 11$

Now, take the square root of both sides:

$w = \sqrt{11} \text{ cm}$

Since $l = 4w$:

$l = 4\sqrt{11} \text{ cm}$

Perimeter Calculation

The perimeter $P$ of a rectangle is calculated as:

$P = 2(l + w)$

Substituting the values of $l$ and $w$:

$P = 2(4\sqrt{11} + \sqrt{11}) = 2(5\sqrt{11}) = 10\sqrt{11} \text{ cm}$

Diagonal Length Calculation

The length of the diagonal $d$ of a rectangle can be found using the Pythagorean theorem:

$d = \sqrt{l^2 + w^2}$

Substituting the values of $l$ and $w$:

$d = \sqrt{(4\sqrt{11})^2 + (\sqrt{11})^2}$

$d = \sqrt{16 \times 11 + 11} = \sqrt{176 + 11} = \sqrt{187} \text{ cm}$

Final Answers

• The perimeter of the rectangle is $10\sqrt{11}$ cm.
• The length of the diagonal of the rectangle is $\sqrt{187}$ cm.

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. If the area of a square is 44 cm², what would be its side length and perimeter?
2. What is the area of a rectangle if its length and width are $2\sqrt{11}$ cm and $\sqrt{11}$ cm, respectively?
3. How would the diagonal change if the width was doubled while keeping the length the same?
4. What is the length of the diagonal of a rectangle whose sides are 7 cm and 24 cm?
5. What is the perimeter of a rectangle whose length is twice its width and has an area of 72 cm²?

Tip: When solving geometric problems, always sketch a quick diagram to better visualize the relationships between the elements like length, width, and diagonal.